If any cloud, LAN, or virtual device is added to a SmartApp, the automation will not run locally. Cloud, LAN, and virtual devices are reliant on an active internet connection. This includes mobile presence devices, as well. Any SmartApps triggered by mobile presence will not execute in the event of an internet outage.

For example:

A Smart Lights SmartApp that controls an GE Link bulb (ZigBee) will run locally.
However, a Smart Lights SmartApp that controls an GE Link bulb (ZigBee) and an LIFX bulb (cloud-to-cloud) will not run locally."

In the context of this requirement, does Google Home (Assistant) count as a “Cloud, Lan, or Virtual Device” ?

Since the installation of Google Home (Assistant) I notice that when I look at My Devices in the IDE, the section titled “In Use By” on a device will include “The Google Assistant” in addition to, for example, “Smart Lighting” and “SmartTiles”. Will this impact whether that device (and hence the Smart Lighting app containing the device) can run locally on the hub?

Will this impact whether that device (and hence the Smart Lighting app containing the device) can run locally on the hub?

No. Where local control is available in SmartApps like SmartLighting, local execution will still work even if you have the device working with Google Home. Google Home will not be able to use local processing because it is cloud to cloud.